What are the quaternion algebras over $\mathbb{F}$ for a field $\mathbb{F}$?

Question

I know that the only quaternion algebras over $\mathbb{R}$ are the quaternions and the split-quaternions. What is the characterization of the quaternion algebras over a particular field? Which of these be extended to octonion algebras?

Answer 1

In general the isomorphism classes of quaternion algebras over a field $K$ are in natural bijection with (among other things):

• the isometry classes of $2$-fold Pfister forms over $K$
• the isometry classes of $3$-dimensional quadratic forms over $K$ of determinant $1$
• the isomorphism classes of projective varieties $V$ over $K$ such that $V$ becomes isomorphic to $\mathbb{P}^2$ over an algebraic closure of $K$
• the cohomology classes in $H^1(K,PGL_2)$.

For a general field there is no neat classification, all those objects can be fairly complicated if the field is complex enough.

Answer 2

There's a cohomological description of certain algebras that is very general and covers in particular quaternion algebras and octonion algebras. I don't know if this is written down anywhere, but the proof is a standard Galois descent argument.

Let $k$ be a field and let $k^s$ be a fixed algebraic closure. Let $A$ be a finite-dimensional algebra over $k$ that is not required to be unital, commutative or associative. Then the group $G$ of $k$-algebra automorphism of $A$ is an algebraic group over $k$. An algebra $B$ (again without any assumptions) over $k$ is called a form of $A$ if $k^s \otimes_k A \cong k^s \otimes_k B$ as algebras over $k^s$. We have the following result:

Forms of $A$ are classified by the Galois cohomology group: $H^1(\operatorname{Gal}(k^s/k),G(k^s))$.

To recover something about quaternion and octonion algebras from this, one can show that forms of any fixed quaternion or octonion algebra are exactly all algebras of that type. Also for quaternion algebras the automorphism group is $PGL_2$, for octonions it's called $G_2$ and is an exceptional algebraic group.

But this result is more general than just these two cases, so one might try to inverstigate, e.g. what happens when one applies it to an algebra over a field $k$ with the same structure constants as the sedenions over $\Bbb R$ (this works as the structure constants are always $0$ or $\pm 1$).

For octonion algebras specifically, they also correspond to Pfister 3-forms, just like quaternion algebras correspond to Pfister 2-forms.